Sven Herrmann
Fachbereich Mathematik Technische Universität Darmstadt
60th Séminaire Lotharingien de Combinatoire April 2, 2008
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 1 / 16
Outline
1 f -Vectors of Simplicial Balls
2 Genocchi Numbers
3 Genocchi Numbers and f -Vectors of Simplicial Balls
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 2 / 16
1 f -Vectors of Simplicial Balls
2 Genocchi Numbers
3 Genocchi Numbers and f -Vectors of Simplicial Balls
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 3 / 16
Simplicial Balls
B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces
boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)
fi(int B) :=fi(B)−fi(∂B)
relation beetween f(∂B)and f(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16
B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces
boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)
fi(int B) :=fi(B)−fi(∂B)
relation beetween f(∂B)and f(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16
Simplicial Balls
B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces
boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)
fi(int B) :=fi(B)−fi(∂B)
relation beetween f(∂B)and f(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16
B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces
boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)
fi(int B) :=fi(B)−fi(∂B)
relation beetween f(∂B)and f(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16
Simplicial Balls
B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces
boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)
fi(int B) :=fi(B)−fi(∂B)
relation beetween f(∂B)and f(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16
B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces
boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)
fi(int B) :=fi(B)−fi(∂B)
relation beetween f(∂B)and f(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16
Simplicial Balls
B an(n−1)-dimensional simplicial ball i.e. simplicial complex homeomorphic to a ball fi(B)number of i-dimensional faces
boundary∂B of B is a simplicial sphere with face numbers fi(∂B) interior int B of B (not a simplicial complex)
fi(int B) :=fi(B)−fi(∂B)
relation beetween f(∂B)and f(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 4 / 16
polytope P: convex hull of finitely many points triangulation T : collection of simplices whose union is P
relation between triangulation of the boundary and T
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 5 / 16
Example: Triangulated Polytopes
polytope P: convex hull of finitely many points triangulation T : collection of simplices whose union is P
relation between triangulation of the boundary and T
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 5 / 16
polytope P: convex hull of finitely many points triangulation T : collection of simplices whose union is P
relation between triangulation of the boundary and T
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 5 / 16
Example: Triangulated Polytopes
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
Example: Triangulated Polytopes
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
Example: Triangulated Polytopes
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
Example: Triangulated Polytopes
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
simplicial polytope (McMullen 2004)
◮ all faces of P are simplices
◮ boundary is a simplicial (triangulated) sphere
◮ f(∂T)is fixed (if we do not allow additional points) triangulations without interior faces of low dimension combinatorics of unbounded polyhedra (H. & Joswig 2007)
◮ complex of bounded faces is dual to interior of a simplicial ball
◮ special case: tight span of finite metric space (Dress 1984, et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 6 / 16
Dehn-Sommerville Relations
hk =Pk
i=0(−1)k−i nn++1−k1−i
fi, g0=1 and gk =hk−hk−1for k ≥1
Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere
hk(S) = hn−k(S) .
Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball
gk(∂B) = hk(B)−hn−k(B) .
from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16
Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere
hk(S) = hn−k(S) .
Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball
gk(∂B) = hk(B)−hn−k(B) .
from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16
Dehn-Sommerville Relations
hk =Pk
i=0(−1)k−i nn++1−k1−i
fi, g0=1 and gk =hk−hk−1for k ≥1
Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere
hk(S) = hn−k(S) .
Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball
gk(∂B) = hk(B)−hn−k(B) .
from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16
Theorem (Dehn-Sommerville) S simplicial(n−1)-sphere
hk(S) = hn−k(S) .
Theorem (McMullen & Walkup 1971) B simplicial(n−1)-ball
gk(∂B) = hk(B)−hn−k(B) .
from there one gets relations beetween h(∂B)and h(int B) (McMullen 2005 et al.)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 7 / 16
Relations for the f -Vector
Proposition (Klain) S simplicial(n−1)-sphere
fk(S) =
n−1
X
i=k
(−1)i+n−1 i+1
k +1
fi(S)
Proposition (Klain) B simplicial(n−1)-ball
fk(int B) = −fk(∂B)
2 −
n−k−1
X
i=1
(−1)n+k+i 2
k+1+i k+1
fk+i(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 8 / 16
S simplicial(n−1)-sphere
fk(S) =
n−1
X
i=k
(−1)i+n−1 i+1
k +1
fi(S)
Proposition (Klain) B simplicial(n−1)-ball
fk(int B) = −fk(∂B)
2 −
n−k−1
X
i=1
(−1)n+k+i 2
k+1+i k+1
fk+i(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 8 / 16
Outline
1 f -Vectors of Simplicial Balls
2 Genocchi Numbers
3 Genocchi Numbers and f -Vectors of Simplicial Balls
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 9 / 16
function
2t et +1 =:
∞
X
n=0
Gntn n! =t+
∞
X
n=1
G2n t2n (2n)! .
The first few numbers are
-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905
One has
Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16
Genocchi Numbers
Definition
TheGenocchi numbersare defined by means of the generating function
2t et +1 =:
∞
X
n=0
Gntn n! =t+
∞
X
n=1
G2n t2n (2n)! .
The first few numbers are
-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905
One has
Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16
function
2t et +1 =:
∞
X
n=0
Gntn n! =t+
∞
X
n=1
G2n t2n (2n)! .
The first few numbers are
-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905
One has
Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16
Genocchi Numbers
Definition
TheGenocchi numbersare defined by means of the generating function
2t et +1 =:
∞
X
n=0
Gntn n! =t+
∞
X
n=1
G2n t2n (2n)! .
The first few numbers are
-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905
One has
Gn =2(1−22n)Bn, where Bnare theBernoulli numbers.
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 10 / 16
Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler
intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &
Zhou 2006)
A001469 in the OEIS
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16
Genocchi Numbers
named after the Italian mathematician Angelo Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler
intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &
Zhou 2006)
A001469 in the OEIS
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16
Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler
intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &
Zhou 2006)
A001469 in the OEIS
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16
Genocchi Numbers
named after the Italian mathematician Angelo Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler
intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &
Zhou 2006)
A001469 in the OEIS
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16
Genocchi (1817-1889) by Edouard Lucas (1891) investigation goes back to Leonhard Euler
intensively studies by Eric T. Bell (1926, 1929) a lot of generalizations (e.g. Kim et al. 2001, Domaritzki 2004, Zeng &
Zhou 2006)
A001469 in the OEIS
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 11 / 16
Combinatorical Interpretation
combinatorial interpretation due to Dominque Dumont (1974):
absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.
n=2:
1 2 3 4 2 1 4 3
1 2 3 4 3 1 4 2
1 2 3 4 3 2 4 1
=⇒ |G6|=3
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16
absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.
n=2:
1 2 3 4 2 1 4 3
1 2 3 4 3 1 4 2
1 2 3 4 3 2 4 1
=⇒ |G6|=3
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16
Combinatorical Interpretation
combinatorial interpretation due to Dominque Dumont (1974):
absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.
n=2:
1 2 3 4 2 1 4 3
1 2 3 4 3 1 4 2
1 2 3 4 3 2 4 1
=⇒ |G6|=3
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16
absolute value of G2n+2equals the number of permutationsτ of {1,2, . . . ,2n}such thatτ(i)>i if and only if i is odd.
n=2:
1 2 3 4 2 1 4 3
1 2 3 4 3 1 4 2
1 2 3 4 3 2 4 1
=⇒ |G6|=3
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 12 / 16
Recursion Formulae
There are a lot of formulae to compute Gn
Proposition
G2n =−n−1 2
n−1
X
k=1
2n 2k
G2k and G2n =−1−
n−1
X
k=1
2n 2k−1
G2k 2k Lemma
For n≥2 we have 2n−1
n G2n =−1 2
n−1
X
k=1
2n−1 2k −1
2k−1 k G2k
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 13 / 16
Proposition
G2n =−n−1 2
n−1
X
k=1
2n 2k
G2k and G2n =−1−
n−1
X
k=1
2n 2k−1
G2k 2k Lemma
For n≥2 we have 2n−1
n G2n =−1 2
n−1
X
k=1
2n−1 2k −1
2k−1 k G2k
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 13 / 16
Recursion Formulae
There are a lot of formulae to compute Gn
Proposition
G2n =−n−1 2
n−1
X
k=1
2n 2k
G2k and G2n =−1−
n−1
X
k=1
2n 2k−1
G2k 2k Lemma
For n≥2 we have 2n−1
n G2n =−1 2
n−1
X
k=1
2n−1 2k −1
2k−1 k G2k
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 13 / 16
1 f -Vectors of Simplicial Balls
2 Genocchi Numbers
3 Genocchi Numbers and f -Vectors of Simplicial Balls
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 14 / 16
Main Theorem
Theorem (H. 2007)
For each simplicial(n−1)-ball B and n−k even we have
fk(int B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B)
−
k +2i k+1
fk+2i−1(int B)
! .
fk for n−k odd can be computed from the even ones
fk(int B)depending an⌊(n−k)/2⌋fi(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 15 / 16
For each simplicial(n−1)-ball B and n−k even we have
fk(int B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B)
−
k +2i k+1
fk+2i−1(int B)
! .
fk for n−k odd can be computed from the even ones
fk(int B)depending an⌊(n−k)/2⌋fi(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 15 / 16
Main Theorem
Theorem (H. 2007)
For each simplicial(n−1)-ball B and n−k even we have
fk(int B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B)
−
k +2i k+1
fk+2i−1(int B)
! .
fk for n−k odd can be computed from the even ones
fk(int B)depending an⌊(n−k)/2⌋fi(int B)
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 15 / 16
n−k even and k ≤e
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i k +1
fk+2i−1(int B)
⌊(e+2)/2⌋
⌊(e+3)/2⌋
equations
⌊(n−1−e)/2⌋(
⌊(n−e)/2⌋
unknowns fk(int B)
similar equations for the h-vector computed and used: H. &
Joswig 2007
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16
Simplicial Balls without Small Interior Faces
B a simplical(n−1)-ball without interior faces of dimension≤e n−k even and k ≤e
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i k +1
fk+2i−1(int B)
⌊(e+2)/2⌋
⌊(e+3)/2⌋
equations
⌊(n−1−e)/2⌋(
⌊(n−e)/2⌋
unknowns fk(int B)
similar equations for the h-vector computed and used: H. &
Joswig 2007
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16
n−k even and k ≤e
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i k +1
fk+2i−1(int B)
⌊(e+2)/2⌋
⌊(e+3)/2⌋
equations
⌊(n−1−e)/2⌋(
⌊(n−e)/2⌋
unknowns fk(int B)
similar equations for the h-vector computed and used: H. &
Joswig 2007
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16
Simplicial Balls without Small Interior Faces
B a simplical(n−1)-ball without interior faces of dimension≤e n−k even and k ≤e
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i k +1
fk+2i−1(int B)
⌊(e+2)/2⌋
⌊(e+3)/2⌋
equations
⌊(n−1−e)/2⌋(
⌊(n−e)/2⌋
unknowns fk(int B)
similar equations for the h-vector computed and used: H. &
Joswig 2007
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16
n−k even and k ≤e
⌊n−2k⌋
X
i=1
G2i 2i
k+2i−1 k+1
fk+2i−2(∂B) =
⌊n−2k⌋
X
i=1
G2i 2i
k+2i k +1
fk+2i−1(int B)
⌊(e+2)/2⌋
⌊(e+3)/2⌋
equations
⌊(n−1−e)/2⌋(
⌊(n−e)/2⌋
unknowns fk(int B)
similar equations for the h-vector computed and used: H. &
Joswig 2007
Sven Herrmann (TU Darmstadt) Genocchi Numbers SLC 60, April 2, 2008 16 / 16